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3
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71 views

Is there a closed-form approximation to a band-limited sawtooth?

A partial Fourier Series with no coefficients is equal to the closed form expression: $${A \over n} \sum_{k=1}^n \cos(k\theta) = {A \over 2n} \left\{{\sin([2n + 1]\theta/2) \over \sin(\theta/2)} - ...
1
vote
1answer
16 views

How to calculate the partition function of a given distribution?

As noted in A FULL BAYESIAN APPROACH FOR INVERSE PROBLEMS, let $ y = Ax + n$, where $x$ is a $m$ dimensional signal and $n$ is white Gaussian noise with precision $\beta$, so we have: $$ y|x, \beta ...
1
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1answer
15 views

Can an arbitrary real function be written in terms of quadratures of an arbitrary frequency with time dependent coefficients?

Given a real function $f$, and a frequency $\Omega$, is it the case that there exist two other real functions $I$ and $Q$ such that $f$ can be written as $$f(t) = I(t) \cos(\Omega t) - Q(t) ...
2
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0answers
28 views

Sampling a Chebyshev polynomial with the discrete cosine transform

I have a Chebyshev polynomial $f$ of degree $n$ in point-value form \begin{align} f&=:S = \left( \left( x_i, y_i \right) \right)_{i=0}^n, \tag{1} \\ x_i &= \cos\left( \frac{i \pi}{n} \right), ...
2
votes
2answers
704 views

Approximate a polynomial function using a sum of sine waves

I have a polynomial function which I need to approximate by a sum of sine waves with constant amplitude along a given domain. From what I hear, this might be a good time to make use of Fourier ...
0
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2answers
63 views

Help needed with the integral of an infinite series

Can you please help me with the integral of this series? I came across it in a signal processing paper and haven't been able to figure out the solution myself. $$ ...
0
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0answers
15 views

DFT of subdomain of periodic domain

$f(t_i,x_j)$ is a solution of stochastic differential equation on grid. $j=[0,N+1]$, $i=[0,\infty]$ and boundary conditions are periodic: $f(t_i,x_0) = f(t_i,x_N)$ and $f(t_i,x_{N+1}) = f(t_i,x_1)$ ...
0
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0answers
28 views

Example of BIBO stable system that is not internally stable

In the theory of system, we know that a system can be BIBO stable but not internally stable (if there is a pole-zero cancellation in the transfer function for example). I find this concept quite ...
1
vote
1answer
775 views

Calculate phase and amplitude of a sampled sine wave

I have an electronics project where I sample two sine waves. I would like to know what the amplitude (peak) and difference in phase is. Actually I just need to know the average product of the two ...
-3
votes
0answers
27 views

Fourier Transform on $L^1(\mathbb{R})$

For $f,g\in L^1(\mathbb{R})$, prove or disprove: $\hat{f}(\xi)+e^{i\pi \xi^2}\hat{g}(\xi) = 0$ for all $\xi\in\mathbb{R}$ implies $\hat{f} = \hat{g} = 0$.
2
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1answer
30 views

Inverse-Fourier transform of a function after non-linear frequency modulation

Suppose $g\in L^1(\mathbb{R})$ such that $\hat{g}\in L^1(\mathbb{R})$ too. So $\tilde{g}(x) = \int_{-\infty}^{\infty}e^{i\pi \xi^2}\hat{g}(\xi)e^{2\pi i \xi x}\,d\xi$ is well-defined. Question is: Is ...
0
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1answer
40 views

Compressive sensing for complex matrix

I'm fairly new to compressive sensing, and I have been looking for a MATLAB implementation of the problem $$ A x = b $$ where $A$ is non square, $x$ is kind of sparse and all the numbers involved are ...
0
votes
0answers
12 views

How to apply a time shift to a pulse-shape, spanned with spline functions?

I have a sampled pulse shape: $ h = [1, 0.5]$ and I do not know what is its real underlying continuous-time pulse. I want to compute the samples of $h(t-\Delta t)$. If I write the continuous pulse ...
2
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0answers
29 views

Generating cross-correlated stochastic processes

I am looking for a robust way to represent and generate multiple stochastic processes that contain time and cross-correlations i.e. I am looking at stochastic processes $X_t^{1}$, $X_t^{2}$, $\ldots$, ...
1
vote
1answer
479 views

How do I compute “AUC” Area under the curve number, if all I have are my TPR and FPR values?

I am trying to rank my neural network, which is trained for binary classification. That is, given a set of input signals, it outputs either a 1 or a 0. I have a training set, where I have the actual ...
0
votes
1answer
52 views

Using Product rule to take derivative

I'm trying to take the derivative of: $$\frac{-1}{6}(e^{-3t}-1) u(t)$$ The $u(t)$ is the step response. So the answer I get is by just doing product rule: ...
0
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2answers
1k views

Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...
3
votes
1answer
140 views

Artifacts and low frequencies FFT.

I am working on analyzing a time signal and want to preform a FFT. However I run in to some artifacts at low frequencies. I have managed to reproduce the behavior in a test signal. Given by $S(t) = ...
1
vote
0answers
22 views

How to decorrelate/Whiten a non-white additive random variable?

I have a signal processing problem where I have the Additive Noise Model (assume Gaussian noise). $$ y = x + w $$ where, $y$ is corrupted signal, $x$ is original signal & $w$ is a non-white ...
3
votes
0answers
32 views

Looking for Math books recommendations to study Electronics

My background is the very basics, and I mean, literally, I can add, sub,mul,div and a little of algebra (near, nothing) and that's it. As you can see I need the best Total Beginner Book(s) that can ...
1
vote
2answers
60 views

How does one verify if a vector is really recovered?

In compressed sensing, how to verify if a vector is really recovered or how does one plot the figures on recovery rate? Since in numerical experiments, there is always a difference between the ...
2
votes
2answers
42 views

Inversion of the Burrows Wheelers Transform

The "Burrows-Wheeler Transform" in signal processing is a transformation which is used in for instance data compression and pattern recognition. It can be described in mathematical terms as: Start ...
1
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2answers
226 views

Can the measurement matrix used for compressive sensing be a sparse matrix?

I am interested in analyzing Compressive Mechanism: Utilizing Sparse Representation in Differential Privacy. In my research, the measurement matrix $A\mathbb \in R^{m \times n}$ needs to be sparse. ...
1
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3answers
41 views

Convert from complex exponentials to sinusoids

I'm working through some notes on signals and systems, and got stuck trying to fill in the missing steps in converting the left hand side to the right hand side of the following equality: $$ \alpha_i ...
0
votes
2answers
426 views

Plot recursive signal in Matlab

I need to create and plot this signal in matlab with 2000 points: x(n) = 0.6530 x(n-1) - 0.7001 x(n-2) + v(n) Where $x(-1)=x(-2)=0$ and $v(n) =$ white noise I ...
0
votes
1answer
23 views

What is the impulse response of the system

Given this input-output system what is the impulse response 𝑑𝑦(𝑡)/dt + 𝑦(𝑡) = 𝑡𝑥(𝑡), 𝑡 ≥ 0, 𝑦(0) = 0 I used an integrating factor to find y(t) y(t) = ${\int t*x(t) *e^tdt\over e^t }$ ...
0
votes
2answers
70 views

What could be the mathematical equation of the given signal?

We know that Fourier series for periodic signal $y(t)$ is given by $$ y(t) = \sum\limits_{m=0}^{+\infty} a_m \cos(w_m t) + \sum\limits_{m=0}^{+\infty}b_m \sin(w_m t). \quad (2)$$ Now,I want to find ...
-1
votes
1answer
50 views

Fourier synthesis of periodic signals

I was reading the Fourier synthesis of periodic signals But I didn't understand the sentence i.e. "Although the calculation of $a_0, a_1, b_1, a_2, b_2$, is a mathematically straightforward ...
0
votes
2answers
51 views

How can I calculate the frequency when my samples don't span a whole period?

I am taking samples at 30Hz of a signal which is a slowly-varying sinewave. The period of the sinewave is expected to drift slowly, but will always be somewhere between 0.5s and 2s. I would like ...
1
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0answers
21 views

Haar wavelet transformation of a binary vector data over $GF(2)$

I am trying to perform Haar wavelet transformation on the following vector which is defined over $GF(2)$. $[1, 0, 1, 0, 1, 0, 1, 0]$ I am doing it as follows. $[1, 0, 1, 0, 1, 0, 1, 0]$ $\implies ...
1
vote
2answers
26 views

particular solution of a difference equation

I am unable to find a particular solution of the following difference equation $$ y[k-1]-5y[k]+6y[k+1]=-u[k-1]+4u[k] $$ with $u[k]=\big(\frac{1}{2}\big)^k$. This is what I tried so far. Because ...
1
vote
2answers
49 views

Kolmogorov-Zurbenko filter - Calculation of coefficients

I'm currently researching the Kolmogorov-Zurbenko filter and trying to implement it myself as a way to smooth one-dimensional signal strength values. The basic filter per se is pretty easy to ...
0
votes
1answer
54 views

Evaluation of the integral $\int_0^1 e^{2t^2 -at} dt$

I would like to integrate a function in the range $[0,1]$. I tried a lot of ways including Mathlab. All solutions come in terms of some error function. I would like the answer in terms of $a$. ...
0
votes
0answers
23 views

Unknown variable in formula - binomial coefficient? [duplicate]

I'm currently researching a filter and don't quite understand one of the equations used there, since it contains a variable I don't know how to calculate: $$(1)\,\,a^{m, k}_s = \frac{c^{k, ...
0
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1answer
1k views

DSP, discrete time, how to calculate the frequency

this is from Digital Signal Processing, 4th ed, Sanjit K Mitra, problem 2.39b. The question is: Determine the fundamental period of $x[n] = cos(0.6n\pi + 0.3\pi)$ Since x[n], is in square brackets, ...
0
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0answers
29 views

deconvolution of exp($x^2$)

I would like to know whether we can get the function of type exp($x^2$) by convoluting any functions. That is which function convolution gives exp($x^2$). Thanks in advance
2
votes
1answer
347 views

Kalman Filter Process Noise Covariance

I want to model the movement of a car on a straight 300m road in order to apply Kalman filter on some noisy discrete data and get an estimate of the position of the car. In a Kalman filter the matrix ...
6
votes
2answers
364 views

Image Reconstruction:Phase vs. Magnitude

Figure 1.(c) shows the Test image reconstructed from MAGNITUDE spectrum only. We can say that the intensity values of LOW frequency pixels are comparatively more than HIGH frequency pixels. $$ ...
1
vote
2answers
69 views

How to derive FWHM of sinc function

So this is probably a simple question, but I am unable to get my head around it. If we have $\operatorname{sinc}(2 \pi v L)$, what is the width of that $\operatorname{sinc}$ in terms of $v$ at half ...
0
votes
1answer
355 views

What is a moving average system?

Can someone elaborate on what a moving average system is? I know that the system is defined as: $$y[n] = \frac{x[n] + x[n-1] + x[n-2]}{3}$$ How would we draw $y[n]$ given that we have a graph with ...
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0answers
18 views

How to analyse data samples scattered in time?

I want to analyse data corresponding to events happening at arbitrary moments in time, and conveying quantitative information. My goal is to study the relationship between the sum of these quantities ...
3
votes
1answer
343 views

Plotting discrete time signals involving sumations in matlab.

Many of the examples I've encountered while looking for an answer are simple functions that do not involve summations. Suppose I have the following function; ...
5
votes
2answers
104 views

Is Fourier series used always for periodic signals and Fourier transform for aperiodic signals only?

I want to ask basic question. In our mathematics classes ,while teaching the Fourier series and transform topic,the professor says that when the signal is periodic ,we should use Fourier series and ...
1
vote
1answer
48 views

What if the Fourier series of a periodic function also has periodic coefficients $a_k$

If given that $x(t)$ is a periodic continuous time signal, with periodic $T$. It can be expressed by the Fourier series, i.e. $x(t)=\sum\limits_{k=-\infty}^{+\infty}\,a_k\cdot e^{j k \frac{2 ...
0
votes
1answer
22 views

Show that a sinusoid having a frequency larger than one corresponds to a sinusoid having a frequency less than one.

I am studying electrical engineering for fun online. There is this one solution to a question on an online textbook that does not make any sense to me. The question is: Show that $\cos(2\pi ...
2
votes
1answer
34 views

Convolution of various functions

There is asked in an example to do convolution $ h_1(t)*h_2(t) + h_3(t)*h_4(t) $ where $h_1(t) = e^{-2t}u(t)$ $h_2(t) = 2e^{-t}u(t) $ $h_3(t) = e^{-3t}u(t) $ $h_4(t) = 4\delta(t) $ and then the ...
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2answers
32 views

Why exponential is ignored in particular solution for impulse response ??

For a system govern by the equation: $$ 2y'(t) +4y(t) =3x(t) $$ To calculate it's impulse response we replace $y(t)$ with $h(t)$ and $x(t)$ with $\delta(t)$ and get $2h'(t)+4h(t)=3\delta(t)$ which's ...
2
votes
0answers
15 views

Rate of convergence of a Weyl-Heisenberg (Gabor) frame expansion

If $\{g_{m,n}\}$ is a Gabor frame for $L^2(R)$, with window function $g$, and $f \in L^2(R)$, is there a bound on the approximation error of $f$ using a finite subset of the frame? That is, is ...
1
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0answers
15 views

Nonparametric changepoint detection for point process

This is a replication of a question I've recently asked on Cross Validated. It hasn't received an answer or much attention, so I've posted it here. I have a family of point processes representing ...
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1answer
46 views

First Order differential equation … How did they do it ?

I was studying "continuous and discrete signals and systems" by Samir S. Soliman where I encountered with this first order differential equation: $$ \frac{dy(t)}{dt} + \frac{R_1R_2}{L(R_1+R_2)}y(t) = ...